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On the invariant subspace problem in Hilbert spaces
Per H. Enflo Available online >> arXiv:2305.15442 [pdf,ps,other]
In this paper we show that every bounded linear operator T on a Hilbert space H has a closed non-trivial invariant subspace. Submitted 24 May, 2023;originally announced May 2023. Comments:13 pages MSC Class:47A15; 47B02
Abstract
A method to construct invariant subspaces for a general operator T on Hilbert space is presented. It represents a new direction of my "method of extremal vectors", first presented in 1998 in Ansari-Enflo [1].
The Main Construction of the method gives a non-cyclic vector of T by gradual approximation by "almost non-cyclic" vectors. There are reasons, why the Main Construction cannot work for some weighted shifts. But when the Main Construction fails, one gets, by using the information obtained, invariant subspaces of T, similar to those of weighted shifts.
The sequence y(n) of "almost non-cyclic" vectors follows the formula y(n+1) = y(n) + r(T) y(n), which will allow for efficient use of elementary Fourier Analysis. The more general formula y(n+1) = y(n) + z would lead to difficult problems concerning the relation between T and T*, problems which may be of interest in themselves.
REFERENCE
Invariant Subspace Problem in Hilbert Spaces:
Exploring Applications in Quantum Mechanics, Control Theory, Operator Algebras, Functional Analysis and Accelerator Physics - Mostafa Behtouei arXiv:2306.17023
"Per H. Enflo’s recent claim of solving the invariant subspace problem in his recent publication titled ”On the invariant subspace problem in Hilbert spaces.” [4] has generated excitement and anticipation in the mathematical community, as his work could represent a major breakthrough in the field. Enflo’s solution, if validated,would be a groundbreaking achievement in mathematics, resolving a longstanding problem and advancing our understanding of bounded linear operators on Hilbert spaces. Its impact would be significant and far-reaching." From: Mostafa Behtouei, 29 Jun 2023
Interdisciplinary Mathematics Institute (IMI)
25 de mayo de 2023.Boletín del IMI.
A potential definitive answer to the invariant subspace problem in Hilbert spaces. Prof. Per H. Enflo (Kent State University, USA) has just announced via his collaborators from the Interdisciplinary Mathematics Institute (IMI) that he might have just potentially solved the long standing Invariant Subspace Problem in Hilbert Spaces
A history of solving some famous problems in mathematical analysis
This paper provides a historical view of what functional analysis is, a biography of Per H. Enflo and a short view of four fundamental problems in the geometry of Banach spaces.
It gives the story of how Enflo, after several years of work, managed to solve three of these problems. It is a story of trying many ideas, testing them on simpler problems, trying to combine different techniques for handling different aspects of the problems, making many failed attempts and finally succeeding.
In 1966 Enflo found a general strategy for solving many problems in infinite dimensions, by combining appropriate finite-dimensional techniques with induction procedures. When applied to the old, fundamental problems in functional analysis, this has led to new concepts, problems, techniques and results in analysis. These developments have been applied also in several other areas of mathematics and computer science.
P H ENFLO
Department of Mathematical Sciences, Kent State University, Kent, OH, USA
M S MOSLEHIAN
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad, Iran
J B SEOANE-SEPÚLVEDA
Instituto de Matemática Interdisciplinar (IMI), Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid, Spain
When did you make the decision to go into math instead of music?
I have never made, and will never make, any such decision
"An infinite-dimensional version of Hilbert's fifth problem with connections to non-linear geometry of Banach spaces"
Lecture December 2021 Slides >> to Holiday Lecture by Per Enflo via Zoom - Jani Virtanen (Reading and Helsinki) & Håkan Hedenmalm (KTH and Reading)Mathematics >> Music >> Interviews & Reviews >>
Per Enflo - mathematician and pianist - was the last laureate of a prize
for solving a scientific problem from the Scottish Book.
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